Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation $$\sum_{d|n}x_d\phi(d)=n$$ where $\phi$ is the Euler's totient function, $d$ is any positive divisor of $n$ and the $x_d$ are the unknowns (see this). Call such a solution a $G$-solution.

Are there number-theoretic conditions which characterize $G$-solutions of the equation where $G$ is a simple group?